Conformal Fitness and Uniformization of Holomorphically Moving Disks

TL;DR

This paper investigates conditions under which a family of holomorphically moving disks admits a holomorphic family of Riemann maps, providing multiple equivalent characterizations and exploring their implications.

Contribution

It establishes five equivalent conditions for the existence of holomorphic Riemann maps for holomorphically moving disks, linking analytic, dynamical, and measure-theoretic perspectives.

Findings

Five equivalent conditions for holomorphic Riemann maps

Connections between analytic, dynamical, and measure-theoretic characterizations

Implications for conformal and uniformization theories

Abstract

Let $\{U_t \}_{t \in {\mathbb D}}$ be a family of topological disks on the Riemann sphere containing the origin 0 whose boundaries undergo a holomorphic motion over the unit disk $\mathbb D$. We study the question of when there exists a family of Riemann maps $g_t:({\mathbb D},0) \to (U_t,0)$ which depends holomorphically on the parameter $t$. We give five equivalent conditions which provide analytic, dynamical and measure-theoretic characterizations for the existence of the family $\{g_t \}_{t \in {\mathbb D}}$, and explore the consequences.